| Summary: | 碩士 === 國立臺灣大學 === 數學研究所 === 103 === The highly oscillatory initial value problem of wave equations has wide applications in science. When the frequency of the wave is very high, traditional numerical methods take much computational time. For this reason, many approximation approaches to this problem are developed. We use an approximation method based on the Liouville equation introduced in the thesis. In this thesis, we use some numerical examples to show that the Hamiltonian-preserving method which is introduced by Jin and Wen [ J. Comput. Phys. 214 (2006), no. 2, 672-697. ] is better than traditional upwind method when the wave speed is discontinuous. We also show that the highly oscillatory energy density of the highly oscillatory initial value problem weakly converges to the Liouville energy density based on the Liouville-quation approach via some examples. In these examples, We determine how high the frequency is, the $ L^1 $-norm of highly oscillatory energy density converges to the $ L^1 $-norm of Liouville energy density. But highly oscillatory energy density does not converge to Liouville energy density in $ L^1 $. Finally, We also give the derivation of solutions of Liouville equaiton with different type of wave speeds.
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